POLITECNICO DI TORINO Analysis of Electric Propulsion Transfers For

Home , Argument of periapsis, Azimuth, Collision, Gridded ion thruster, Mean anomaly, Orbital elements

POLITECNICO DI TORINO

Dipartimento di Ingegneria Meccanica e Aerospaziale

Corso di Laurea Magistrale in Ingegneria Aerospaziale

Tesi di Laurea Magistrale

Analysis of Electric Propulsion transfers for multiple debris removal

Relatore Candidato prof. Lorenzo Casalino Biagio Cotugno

Marzo 2018

Summary

1 Introduction 1

2 Debris and debris removal 5

3 Mathematical model 10

3.1 Orbit perturbations 10

3.2 Transfer description for impulsive manoeuvres 15

3.3 Corrections to the algorithm for Electric Propulsion 21

4 Electric Propulsion 26

4.1 History background 26

4.2 Fundamentals of Electric propulsion 27

4.3 Thrusters classification 31

4.4 Propellants 47

5 Results 48

5.1 Output tables with dimensional results 50

5.2 Mass budget 61

5.3 Electric propulsion – Chemical propulsion comparison 63

6 References 70

7 Acknowledgments 71

Index of figures

Figure 1: Space debris population ...... 6

Figure 2: Kessler's syndrome trend ...... 7

Figure 3: Frequency of ISS collision avoidance manoeuvres between 1999 and 2014 (Image credit: NASA) ...... 9

Figure 4: Effect of perturbations forces on orbital elements ...... 10

Figure 5: Satellite orbit and orbital elements ...... 13

Figure 6: Ascending node variation due to J2 perturbations on 4 debris ...... 13

Figure 7: Ascending node variation (related to Debris1) due to J2 perturbations on 4 debris ...... 14

Figure 8: Orbital elements of target set (Epoch 57754 Modified Julian Date) ..... 15

Figure 9: Performance of three-leg sequences for the 82-deg cluster ...... 16

Figure 10: Performance of three-leg sequences for the 82-deg cluster ...... 16

Figure 11: Performance of three-leg sequences for the 82-deg cluster ...... 17

Figure 12:Electric propulsion window manoeuvre for thrust time lower than time necessary to have coincident orbit planes ...... 22

Figure 13: Electric propulsion windows manoeuvre for thrust time greater than the time necessary to have coincident orbit planes but lower than twice it ...... 23

Figure 14:Electric propulsion window manoeuvre for thrust time greater than twice the time necessary to have coincident orbit planes ...... 24

Figure 15: SERT-1 ...... 26

Figure 16:Zond-2 ...... 26

Figure 17: Rocket engine thrust ...... 28

Figure 18: Arcjet ...... 33

Figure 19: Arcjet,푉 푣푠푙푛푗 ...... 33

Figure 20: Diagram of a Gridded Ion Thruster ...... 36

Figure 21: Hall thruster ...... 39

Figure 22: MPD thruster ...... 42

Figure 23: VASIMR ...... 44

Figure 24: Pulsed Inductive Thruster (PIT) ...... 46

Figure 25: Propellant Injection Schematic ...... 46

Figure 26:휴 perturbation for the debris combination over the mission time ...... 49

Figure 27: 2D Graphic (Ground track) of the four-debris ...... 59

Figure 28: 3D Graphic (Orbit View) of the four-debris ...... 59

Figure 29: Spacecraft mass decrease during the best three-leg mission for CP and EP thruster, for all the acceleration values ...... 64

Figure 30: Propellant mass consumption for the best three-leg mission for CP and EP thruster, for all the acceleration values ...... 65

Figure 31: Power depending on Solar panel mass for the three electric thrusters 66

Figure 32: Propellant mass depending on Solar panel mass for the three electric thrusters ...... 67

Index of tables

Table 1: Earth flattening...... 11

Table 2: Momentum conservation law terms ...... 27

Table 3: Thrust terms ...... 27

Table 4: Specific impulse law terms ...... 29

Table 5: Tsiolkovsky rocket equation terms ...... 30

Table 6: Resistojets performance ...... 34

Table 7: Arcjets performance ...... 35

Table 8: Hall thrusters vs Ion thrusters performance ...... 40

Table 9: PPT vs AF-MPD vs SF-MPD performance ...... 43

Table 10: List of typical propellants for EP ...... 47

Table 11: Thrust on mass ratios (Acceleration) ...... 50

Table 12: Solar panel mass and Power for 푎 = 1 ⋅ 10 − 5 푚/푠2 ...... 51

Table 13: Solar panel mass and Power for 푎 = 1 ⋅ 10 − 6 푚/푠2 ...... 51

Table 14: Solar panel mass and Power for 푎 = 1 ⋅ 10 − 7 푚/푠2 ...... 52

Table 15: Mean thrust time for different acceleration values ...... 53

Table 16: Best Three-Leg Missions in Terms of 휟푽 for 풂 = ퟏ풆 − ퟎퟔ 풎/풔ퟐ ..... 54

Table 17: Best Three-Leg Missions in Terms of 휟푽 for 풂 = ퟏ풆 − ퟎퟓ 풎/풔ퟐ ..... 54

Table 18: Best Three-Leg Missions in Terms of 휟푽 for 풂 = ퟏ풆 − ퟎퟕ 풎/풔ퟐ ..... 55

Table 19: Best Three-Leg Missions in Terms of Trip Time 휟풕 for 풂 = ퟏ풆 − ퟎퟔ 풎/풔ퟐ ...... 55

Table 20: Best Three-Leg Missions in Terms of Trip Time 휟풕 for 풂 = ퟏ풆 − ퟎퟓ 풎/풔ퟐ ...... 56

Table 21: Best Three-Leg Missions in Terms of Trip Time 휟풕 for 풂 = ퟏ풆 − ퟎퟕ 풎/풔ퟐ ...... 56

Table 22: Minimum delta V for Electric Propulsion based engines ...... 57

Table 23: Maximum delta V for Electric Propulsion based engines ...... 57

Table 24: Four-debris combination providing minimum delta V ...... 57

Table 25: Four-debris combination providing maximum delta V ...... 58

Table 26: Four-debris combination providing minimum mission time ...... 60

Table 27: Minimum mission time for Electric propulsion based engines ...... 60

Table 28: Mass budget breakdown for 풂 = ퟏ ⋅ ퟏퟎ − ퟓ 풎/풔ퟐ ...... 61

Table 29: Mass budget breakdown for 풂 = ퟏ ⋅ ퟏퟎ − ퟔ 풎/풔ퟐ ...... 62

Table 30: Mass budget breakdown for 풂 = ퟏ ⋅ ퟏퟎ − ퟕ 풎/풔ퟐ ...... 62

Table 31: Comparison between EP and CP in terms of 휟푽 ...... 63

Table 32: Comparison between EP and CP in terms of 풕 ...... 63

Table 33: Propellant mass and solar panel mass for EP, 푎 = 1 ⋅ 10 − 5 푚/푠2 ... 68

Table 34: Propellant mass and solar panel mass for EP, 푎 = 1 ⋅ 10 − 6 푚/푠2 ... 68

Table 35: Propellant mass and solar panel mass for EP, 푎 = 1 ⋅ 10 − 7 푚/푠2 ... 69

1 Introduction

La copiosa presenza nello spazio di detriti o “spazzatura” – come meno elegantemente viene di solito indicato tutto quel complesso di frammenti dispersi nello spazio, creati dall’uomo ma che a quest’ultimo non servono più – è un problema attuale che vede coinvolti da anni gli scienziati delle agenzie spaziali di tutto il mondo: un complesso di scarti potenzialmente pericolosi per le successive rotte di osservazione spaziale, in particolar modo in orbita LEO.

Tale problematica, sempre più all’ordine del giorno, è conseguente e strettamente collegata all’aumento esponenziale delle missioni spaziali dopo il lancio di Sputnik 1 che ha portato in orbita una quantità sempre maggiore di “space junk” come ultimi stadi di lanciatori, parti di satelliti o satelliti non più operanti, oltre che da detriti generati da collisioni nello spazio: tutto questo insieme di “immondizia spaziale” (termine preso a prestito dalle relazioni dei più autorevoli scienziati) cosituisce un ostacolo ed un concreto rischio sia per il lancio di nuovi satelliti che per la sicurezza della crew a bordo di uno spacecraft per una missione con equipaggio.

La propulsione elettrica rappresenta una nuova frontiera per lo spazio ed una valida alternativa all’uso a bordo della propulsione chimica. In tale prospettiva notevoli passi avanti sono stati fatti dopo i primi studi eseguiti dallo scienziato statunitense Robert H. Goddard (1882-1945) e dopo i primi esperimenti a bordo del SERT-1 che hanno portato i propulsori elettrici a poter essere utilizzati anche per missioni a breve termine nonostante la produzione della loro spinta sia modesta in quanto diretta conseguenza della potenza generabile a bordo (principalmente per mezzo di pannelli solari).

In questa lavoro viene presentato uno studio di missione che ha per oggetto la rimozione di multiple debris in orbita bassa attorno alla terra, sfruttando un chaser equipaggiato con propulsione elettrica ed un kit operante con propulsione chimica.

Lo studio è stato sviluppato sfruttando un algoritmo ed un modello matematico per la risoluzione dei problemi di Keplero e Lambert che - a partire dalla definizione delle variabili in gioco, dei valori iniziali e finali delle condizioni al contorno - andasse iterativamente ad annullare l’errore tra:

▪ la posizione calcolata del chaser rispetto al debris di arrivo ▪ la posizione ottenuta tenendo conto delle perturbazioni secolari agenti sul nodo ascendente e sull’argomento del periastro dell’orbita dello spacecraft.

L’analisi tende ad esaminare, in primo luogo, tutte le possibili sequenze caratterizzate da un tempo di missione non superiore a 6 mesi (180 giorni) e da un consumo - in termini di Δ푉 - inferiore ad 1.5 km/s, arrivando ad eliminare tutte quelle sequenze considerate “errate” (ossia caratterizzate da passaggi multipli per lo stesso debris o quelle con tempo di rendez-vous tra un detrito e l’altro inferiore al tempo necessario per effettuare la manovra con propulsione elettrica).

Nel presente studio tali sequenze, calcolate per diversi valori di accelerazione dello spacecraft, sono state in via successiva ottimizzate sia in termini di tempo di missione che di consumo di propellente ed i relativi risultati sono stati rapportati e confrontati con quelli ottenuti attraverso una medesima missione effettuata, però, con la propulsione chimica.

È stato, ancora, sviluppato un mass budget per le varie tipologie di propulsori oggetto di studio che ha dovuto tener conto della potenza necessaria al funzionamento del propulsore elettrico, per poi comparare le differenze ottenute con la stessa missione effettuata con propulsore chimico, evidenziandone i rispettivi dati positivi e di criticità.

Le sequenze simili, infine, sono state ulteriormente selezionate ed equiparate tra loro per poter eseguire una valutazione obiettiva sui differenti risultati al fine di individuare il miglior processo di rimozione dei detriti in termini di tempo di missione e di consumo di propellente.

The copious presence of debris in space or "garbage" - as less elegantly it is usually indicated all the complex of fragments dispersed in space, created by man but that is no longer needed by the man - is a current problem that involves scientists from space agencies all over the world: a complex of potentially dangerous waste for the subsequent space observation routes, in particular in LEO orbit.

This problem is still commonplace and is consequent and closely linked to the exponential increase in space missions after the launch of Sputnik 1 which has brought an ever-increasing amount of space junk into orbit as the last stages of launchers, parts of satellites or no longer operating satellites, as well as debris generated by collisions in space: all this set of "space garbage" (term borrowed from the reports of the most authoritative scientists) constitutes an obstacle and a concrete risk for both the launch of new satellites for the safety of the crew on board a spacecraft for a manned mission.

Electric propulsion represents a new frontier for space and a valid alternative to the use on board of chemical propulsion. In this perspective, considerable progress has been made after the first studies carried out by the American scientist Robert H. Goddard (1882-1945) and after the first experiments on board the SERT-1 that led the electric propulsion to be used for short missions too, despite the production of their thrust is modest as a direct consequence of the power that can be generated on board (mainly with solar panels).

In this paper, a mission study is presented that deals with the removal of multiple debris in a low orbit around the earth, exploiting a chaser equipped with electric propulsion and a kit operating with chemical propulsion.

The study was developed using an algorithm and a mathematical model for solving Kepler and Lambert problems that - starting from the definition of the variables involved, the initial and final values of the boundary conditions - was run iteratively to nullify the error between:

▪ the calculated position of the chaser with respect to the arrival debris ▪ the position obtained taking into account the secular perturbations acting on the ascending node and on the argument of periapsis of the spacecraft orbit.

The analysis tends to examine, in first place, all the possible sequences characterized by a mission time not exceeding 6 months (180 days) and by a consumption - in terms of ΔV - less than 1.5 km /s, eventually deleting all those “incorrect” sequences (i.e. characterized by multiple passages for the same debris or those with a time of rendezvous between a debris and another shorter than the time necessary to perform the manoeuvre with electric propulsion).

In the present study, these sequences, calculated for different acceleration values of the spacecraft, were subsequently optimized both in terms of mission time and propellant consumption and the relative results were compared and matched with those obtained through the same mission carried out with the chemical propulsion.

A mass budget has been developed too, taking into account the necessary power for the electric propulsion engines, to compare the differences obtained with the same mission carried out with a chemical engine, highlighting the respective positive and critical data.

Finally, similar sequences have been further selected and compared to each other in order to perform an objective evaluation of the different results to identify the best debris removal process in terms of mission time and propellant consumption.

2 Debris and debris removal

More than 20.000 tons of natural (dust, meteoroids, micrometeoroids, chunks of asteroids, comets) and man-made objects hit the Earth every year.

Meteoroids are in orbit around the sun, while most artificial debris are in orbit around the Earth. Hence, the latter is more commonly referred to as orbital debris.

With every mission, parts of satellites, satellites, pieces of old exhausted booster segments and even astronauts’ equipment (such as astronaut Ed White’s glove from first extra-vehicular activity of 1965 Gemini 4 flight or Gene Rodenberry’s ashes, creator of the Star Trek series) have remained in space so that the environment around Earth is filling up with “junk” which constitutes an increasing risk for manned and unmanned space missions.

Even tiny paint flecks can damage a spacecraft when traveling at these velocities (34.500 km/h). The fused-silica and borosilicate-glass window of the ISS cupola has been damaged for a 7-mm wide dent. In fact, a number of space shuttle windows have been replaced because of damage caused by material that was analysed and shown to be paint flecks.

Current estimations say that there are over 170 million debris smaller than 1cm, 670.000 debris between 1 cm and 10 cm and 29.000 debris larger than 10 cm orbiting around Earth. Among these, 16.000 are man-made non-operational objects that have been tracked by the US Strategic Command.

Collision risks are divided into three categories depending upon size of threat. For objects 10 centimetres and larger, conjunction assessments and collision avoidance manoeuvres are effective in countering objects which can be tracked by the Space Surveillance Network. Objects smaller than this usually are too small to track and too large to shield against. Debris shields can be effective in withstanding impacts of particles smaller than 1 centimetre.

Figure 1: Space debris population

According to Donald J. Kessler (1940 - ) the scenario of space debris in low earth orbit is so dense of debris that the number of collisions grow exponentially every year so that new debris born from their impact, increasing the risk of new collisions (chain reaction) and making future space missions harder to accomplish.

Relative velocities in low orbits are up to 57.600 km/h (16 km/s) so that the kinetic energy of the collision between two “big enough” objects (diameter larger than 10 cm) creates a cloud of debris in the form of splinters launched in random directions.

Each fragment therefore has the potential to induce further impacts, creating an even greater number of spatial debris. With a large collision (for example between a space station and a non-operative satellite), the amount of debris could be sufficient to make the low orbit level virtually unattainable.

However, the Kessler syndrome’s risk does not completely apply to debris in lower orbits because of the atmospheric drag: the resistance they encounter gradually lowers their altitude until they burn in the atmosphere, thus keeping the area clear of waste. Collisions that occur below this altitude are not a problem, since the loss

of energy in the collision causes the debris orbit to have a perigee below that altitude.

At higher altitudes, where the atmospheric drag is not so significant, the time the debris stay in orbit around Earth is considerable. The influence of the moon, solar wind and weak atmospheric drag slowly lower debris orbits up to where they naturally burn but this process can take millenniums to occur.

Figure 2: Kessler's syndrome trend

To reduce the generation of additional space debris, countermeasures can be taken into account:

- Passivation of the spacecraft, i.e. expulsion of excess fuel once the operations have been completed to eliminate (or at least reduce to a minimum) the risk of explosions in orbit which would generate a huge amount of debris. - In the event that the deorbit is too expensive, it could lead the satellite to orbit at a lower altitude so that the deorbit of the same occurs naturally (passive deorbit) in a maximum of 15 years (according to ESA specifications).

- Introduction of parking orbits (graveyard orbits) for satellites orbiting too high to carry out active or passive deorbit (e.g. for satellites operating in geostationary or geosynchronous orbits). - Debris removal with specific missions.

For what concerns the plan for reacting to debris, NASA has developed a set of long-standing guidelines used to assesses whether the treat of a close passage to the debris is sufficient to warrant evasive manoeuvres to ensure the safety of the spacecraft and the crew (in case of a manned mission).

“Debris avoidance manoeuvres are planned when the probability of collision from a conjunction reaches limits set in the space shuttle and space station flight rules. If the probability of collision is greater than 1 in 100.000, a maneuver will be conducted if it will not result in significant impact to mission objectives. If it is greater than 1 in 10.000, a manoeuvre will be conducted unless it will result in additional risk to the crew.

Debris avoidance manoeuvres are usually small and occur from one to several hours before the time of the conjunction. Debris avoidance manoeuvres with the shuttle can be planned and executed in a matter of hours. Such manoeuvres with the space station require about 30 hours to plan and execute mainly due to the need to use the station’s Russian thrusters, or the propulsion systems on one of the docked Russian or European spacecraft.

Several collision avoidance manoeuvres with the shuttle and the station have been conducted during the past 10 years.

NASA implemented the conjunction assessment and collision avoidance process for human spaceflight beginning with shuttle mission STS-26 in 1988. Before launch of the first element of the International Space Station in 1998, NASA and DoD jointly developed and implemented a more sophisticated and higher fidelity conjunction assessment process for human spaceflight missions.

In 2005, NASA implemented a similar process for selected robotic assets such as the Earth Observation System satellites in low Earth orbit and Tracking and Data Relay Satellite System in geosynchronous orbit.

In 2007, NASA extended the conjunction assessment process to all NASA manoeuvrable satellites within low Earth orbit and within 124 miles (200 kilometres) of geosynchronous orbit.

DoD’s Joint Space Operations Center (JSpOC) is responsible for performing conjunction assessments for all designated NASA space assets in accordance with an established schedule (every eight hours for human spaceflight vehicles and daily Monday through Friday for robotic vehicles). JSpOC notifies NASA (Johnson Space Center for human spaceflight and Goddard Space Flight Center for robotic missions) of conjunctions which meet established criteria.

JSpOC tasks the Space Surveillance Network to collect additional tracking data on a threat object to improve conjunction assessment accuracy. NASA computes the probability of collision, based upon miss distance and uncertainty provided by JSpOC.

Based upon specific flight rules and detailed risk analysis, NASA decides if a collision avoidance manoeuvre is necessary.

If a manoeuvre is required, NASA provides planned post-manoeuvre orbital data to JSpOC for screening of near-term conjunctions. This process can be repeated if the planned new orbit puts the NASA vehicle at risk of future collision with the same or another space object.” [Ref. NASA]

Figure 3: Frequency of ISS collision avoidance manoeuvres between 1999 and 2014 (Image credit: NASA)

3 Mathematical model

3.1 Orbit perturbations

Ideal Keplerian orbits assume that the motion of a spacecraft (in general of a body) in its orbit is the result of the attraction between two bodies only.

There are nonconservative perturbing forces, such as solar pressure that tend to change the eccentricity (푒) of the spacecraft orbit or the atmospheric drag that tend to decrease the major axis (푎) of the orbit causing the satellite to fall back to the earth’s surface.

These perturbations can be classified on how they affect the Keplerian elements:

▪ secular variations coincide with a linear variation of the orbital elements ▪ short-period variations that tend to repeat periodically with a period less than the orbit period ▪ long-period variations are the perturbations that repeat periodically with a period greater than the orbit one.

Figure 4: Effect of perturbations forces on orbital elements

The change in any orbital element, c, is here illustrated. The straight line shows the secular effect. The large oscillating line shows the secular plus long-period effects and the small oscillatory line, which combines all three, shows the short-period effects.

The perturbations due to the oblateness of the Earth are the results of assuming that the Earth is not perfectly spherically symmetrical or homogenous in mass: in fact, the Earth has a bulge on the equator and is crushed at the poles.

Table 1: Earth flattening

Earth equatorial radius (푅) 6378.388 km

Earth polar radius (푅′) 6356.912 km

Earth flattening (푅 − 푅′)/푅 = 0.003

The potential generated by the oblateness of the Earth causes both secular and periodical perturbation.

By using the zonal coefficient 퐽푛 for defyning the potential function of the Earth, the results of the non-spherical Earth take into account the 퐽2 coefficient in the geopotential expansion.

Secular variations due to J2 affect only the longitude of the ascending node (Ω), the argument of perigee (휔) and the mean anomaly (M). These variations are the results of the gyroscopic precession of the orbit around the ecliptic pole.

The rates of change of longitude of ascending node (Ω) and argument of periapsis (휔) are as follows:

3 푛퐽 cos(푖) 푅 2 Ω = − 2 ( 푒) 퐽2 2 (1 − 푒2)2 푎

3 푛퐽 [1 − 5 cos2 푖] 푅 2 휔 = − 2 ( 푒) 퐽2 4 (1 − 푒2)2 푎

where n is the mean motion in degrees/day, 퐽2 has the value of 0.001082629, 푅푒 is the Earth equatorial radius of 6378.1363 km, 푎 is the sami-major axis in km, 푖 is the inclination in degrees, 푒 is the eccentricity and Ω and 휔 are in degrees/day.

Before launching the analysis, the perturbation of the Right Ascension of the Ascending Node (RAAN – Ω [deg]), in terms of variation of RAAN over time (Ω̇ ), has been studied.

A perturbation is a deviation from normal or expected motion (deviation from expected orbital elements). When analysing the universe from an accurate point of view, one can distinct, sometimes unpredictably, some irregularities of motion upon the mean motion of celestial bodies.

Some of the perturbations that can be considered are mainly due to attracting bodies, atmospheric drag and lift, asphericity of celestial bodies (e.g. the Earth), solar effects (radiation, magnetism).

For this mission scenario, the perturbation of the orbital elements over time is a problem related to the Earth flattening only (because of the low orbit of the chaser satellite) and has been solved by evaluating the daily perturbation of the RAAN parameter. Ignoring this effect would cause us to completely fail in the prediction of the satellite position over time.

The variation of the Ω parameter directly influences the geometry of the transfer manoeuvre and the Δ푉 required to the thruster (consumption): the optimal strategy is to wait for the ΔΩ – between the two debris – to be null so that no plane change is required but the manoeuvre can be accomplished with a planar transfer.

Figure 5: Satellite orbit and orbital elements

Figure 6: Ascending node variation due to J2 perturbations on 4 debris

Figure 7: Ascending node variation (related to Debris1) due to J2 perturbations on 4 debris

3.2 Transfer description for impulsive manoeuvres

Figure 8: Orbital elements of target set (Epoch 57754 Modified Julian Date)

A population of debris with similar inclination represents the set of possible targets for the space mission.

In this scenario, Russian Kosmos 3M rocket bodies – which are a consistent part of large debris population in low earth orbit (LEO) – have been considered.

The United States Strategic command orbital objects database lists more or less 300 Kosmos exhausted rocket bodies orbiting in LEO (listed with the US classification SL-8 R/B).

Starting from identifying two clusters with inclination close to 74 degrees (120 objects) and 82 degrees (156 objects), this study evaluated the cluster of objects orbiting at 82 degrees of inclination.

Departure date is on 1st January 2017: the TLE (two-line elements) of the cluster are first propagated until departure date and then, their orbits are propagated taking into account ideal Keplerian orbits and secular perturbations.

Relevant orbital elements are represented in Figure 8.

Results show that the specific optimal sequences depend strongly on departure date but the general trends can be assumed for any departure date.

Figure 9: Performance of three-leg sequences for the 82-deg cluster

Figure 10: Performance of three-leg sequences for the 82-deg cluster

Figure 11: Performance of three-leg sequences for the 82-deg cluster

A complete scan of all the suitable mission opportunities taking into account the variability of departure date and arrival time would be almost impossible.

The global search provides suitable sequences of targets and estimation for rendezvous times and mission cost. It assumes that rendezvous occurs when the orbit planes become coincident and estimates the Δ푉 as a function of changes in semimajor axis and eccentricity.

The actual mission feasibility and characteristics have been verified by means of an evolutionary algorithm developed at Politecnico di Torino used to obtain an optimal solution and so minimum Δ푉 for each transfer.

The results review provides reliable values with errors of 1-2 days for the times and a few m/s for the Δ푉.

It is assumed that the i-th leg begins at the time 휏푖 = 푡푖 which derives from the solution of the previous leg and takes into account the moment the orbit planes of the two debris (debris 푖 and debris 푖 + 1) coincide and the servicing time.

The times at which performing thrust are lower than the times 휏1 < 휏2 < 휏3 < 휏4 =

푡푖+1 for each leg. The rendezvous time with the last debris 휏4 = 푡푖+1 is an unknown factor of the problem and its value can only be estimated.

Each trip from one debris to another is a problem limited to 10 variables. Three of these define the moments at which performing thrust:

푝1 = 휏4 − 휏1, 푝2 = (휏2 − 휏1)/(휏4 − 휏1), 푝3 = (휏3 − 휏2)/(휏4 − 휏2)

The flight time 푝1 varies in a 10-day window centered in the one suggested by the global search (20 days are used for the longest trip), 푝2 and 푝3 vary between 0 and 1.

Six variables define the velocity components after the first (marked with the subscript 1+) and the second (marked with the subscript 2+) burn, from the unknown velocity components before these burns (marked with the subscripts 1- and 2- respectively).

푢1+ = 푢1− + 푝4 sin(푝5)

푣1+ = 푣1− + 푝4 cos(푝5) cos(휓 + 푝6)

푤1+ = 푤1− + 푝4 cos(푝5) cos(휓 + 푝6)

푢2+ = 푢2− + 푝7 sin(푝8)

푣1+ = 푣2− + 푝7 cos(푝8) cos(휓 + 푝9)

푤2+ = 푤2− + 푝7 cos(푝8) cos(휓 + 푝9)

−1 휓 = tan (푤−/푣−)

where u, v and w are the radial, eastward and northward velocity components, and 휓 is the heading angle before burn.

푝4 and 푝7 are the speed change components Δ푉1 and Δ푉2 that vary between 0 and

800 m/s. 푝5 and 푝8 are in-plane angles ranging between -180 and 180 deg. 푝6 and

푝9 are out-of-plane angles ranging from -60 to 60 deg.

Both Δ푉 and out-of-plane angles are values close to zero (planar trajectories are preferred but a margin on out-of-plane maneuvers must still be considered) to improve the accuracy and convergence of the algorithm.

The last variable 푝10 refers to the last transfer, to perform the rendezvous with the 푖 + 1 debris that has a definite position.

For this reason, the Lambert problem must be solved. In this problem multiple revolutions are foreseen and the problem therefore presents two solutions: 푝9 is used to choose between the solution of the left branch (푝10 < 0.5) and that of the right branch (푝10 > 0.5).

Once defined the variables of the problem, Δ푉 of the mission can be evaluated. Initial position, initial time and initial velocity (defined with subscript 1-) are known. Once the speed at the end of the burn is known the Kepler problem can be solved (taking into account the perturbation 퐽2) to evaluate and calculate position and velocity before the following manoeuvre (2-).

The same procedure is applied to the next arc, allowing to know the position and velocity before the following transfer.

The transfer between point 3 and point 4 (the target debris whose position and velocity are known) is first solved as an unperturbed Lambert problem to obtain the velocity components after the third impulse that would allow to intercept the arrival debris without taking the perturbations into account.

The perturbations 퐽2 influence the result and therefore an iterative calculation is necessary to nullify the error of the perturbed final position at time 푡4 between chaser and target. Total Δ푉 is:

4 4 2 2 2 Δ푉 = ∑ Δ푉푗 = ∑ √(푢푗+ − 푢푗+) + (푣푗+ − 푣푗+) + (푤푗+ − 푤푗+) 푗=1 푗=1

where the values 1- and 4+ are those of the departure and arrival debris.

Each arc is solved in sequence, starting with the values obtained at the end of the previous one.

It is important to note that the optimal strategy for favourable opportunities is often to wait on the initial orbit for a rather long time (i.e. Δ푉1 = 0), until the orbital planes of the two debris coincide, thanks to the perturbations J2.

At this moment the spacecraft performs an impulsive manoeuvre towards the following debris.

3.3 Corrections to the algorithm for Electric Propulsion

Changes were necessary in the algorithm and in the mathematical model, to take into account the use of electric propulsion.

The type of propulsion (chemical or electric) directly influences preliminary results for consumption (propellant mass and Δ푉), mass budget and mission time: the starting point of the analysis was to properly write down the characteristic velocity requirement for small changes of quasi-circular orbits.

Δ푉 Δ푎 = 0.5 푉 푎 0 0 Δ푉 = 0.649 Δ푒 { 푉0 and by combining the two formulas:

1 Δ푉 = √ [0.5√푎2 + 0.649√Δ푒2] 푎푚푖푛 that represents the formula used in the calculation of the Δ푉 for the mission with electric propulsion.

The “thrust time” (푡푡) has been introduced as an additional parameter of the problem, since the thrust generated by electric propulsion is not impulsive but is defined in a time interval.

The thrust time will modify (and eventually reduce) the possible sequences of debris to be removed because it defines a further comparison parameter in the mission time: if the thrust time is greater than the time necessary to have coincident orbit planes between two debris, the solution will be discarded as it is impossible to perform.

In addition, it is necessary to define a manoeuvre window to univocally identify the temporal instant (defined 푡0) in which to start the thrust manoeuvre useful to the rendezvous, so that at the end of the thrust (time 푡1) the chaser has correctly performed the rendezvous with the desired debris.

The manoeuvre with electric propulsion is not impulsive but lasts for a certain time span defined by the parameter “thrust time” (푡푡). It is impossible to wait for the orbit planes to coincide (ΔΩ = 0) before starting performing the finite manoeuvre.

During the manoeuvre, the spacecraft leaves the orbit of debris 1 (characterized by

Ω1) and enters the orbit of debris 2 (characterized by Ω2). It is supposed that the variation of the Ω parameter (Ω̇ ) is the average between those of the two orbits.

It is desirable that the manoeuvre is centred at the point of coincidence of the orbital planes. There are two relevant cases:

Figure 12:Electric propulsion window manoeuvre for thrust time lower than time necessary to have coincident orbit planes

Thrust time is lower than the time required for the orbit planes to coincide (defined as 푡푥 and so 푡푡 < 푡푥).

The new rendezvous time, taking into account that at time 푡푥 the spacecraft is still performing the manoeuvre, is defined as:

1 푡푡 푡1 = 푡푥 (1 + ) 2 푡푥

Figure 13: Electric propulsion windows manoeuvre for thrust time greater than the time necessary to have coincident orbit planes but lower than twice it

If the time for the thrust requires more than 푡푥 but less than 2푡푥, it is necessary to evaluate a waiting time before starting the rendezvous operation in order to have the centre of the manoeuvre when the orbital planes coincide.

That is to say that the value of Ω1 − Ω2 is strictly dependant on the waiting time

(푡푤), the thrust time (푡푡) and 푡푥.

In fact:

Ω̇ − Ω̇ ΔΩ = Ω − Ω = 푡 ( 1 2) + 푡 (Ω̇ − Ω̇ ) = 푡 (Ω̇ − Ω̇ ) = 0 1 2 푡 2 푤 1 2 푥 1 2

2푡 − 푡 푡 = 푥 푡 푤 2

Or in general:

푡 푡 = 푡 + 푡 = 푡 + 푡 1 푤 푡 푥 2

Figure 14:Electric propulsion window manoeuvre for thrust time greater than twice the time necessary to have coincident orbit planes

Instead, if the thrust time is greater than twice the time necessary for the orbit planes to coincide (푡푡 > 2푡푥), it is impossible to perform a manoeuvre without a plane change.

The solutions are discarded because there’s less time than the necessary one to carry out the rendezvous (the waiting time appears to be negative).

Before compiling the code and running the executable (*.exe), preliminary considerations have been made:

▪ Different types of electric propulsion based engines have been selected with

different specific impulses (퐼푠푝) values: Hall thrusters, ion thrusters and arcjets.

▪ A debris database has been created based on the North America Aerospace Defence Command (NORAD) Catalog Number, containing all the orbital elements (TLE – two-line elements) of n=154 orbiting debris (such as exhausted launchers upper stage) at an inclination of 82 degrees dating back to late 2017.

▪ Fundamental parameter for the comparison with the chemical propulsion (CP) was the Thrust time because the electric propulsion (EP) requires a finite burn contrary to the chemical propulsion provided by a finite one.

▪ A 20-day servicing time (DT) was selected, i.e. enough time for scientific operations such as observation and hooking of the debris.

▪ J2 perturbations have been considered (in other words it was only considered the effect of the Earth flattening on the spacecraft orbiting in LEO). The variation of the Ω parameter over time influences the time of initial burn with electric propulsion. Since the burn is a finite one (and not an impulsive shot) the moment of ignition is certainly earlier than the moment of the impulsive thrust for chemical propulsion.

▪ i=3 legs have been selected, that is to say, a number of debris m=4 has been chosen providing 154!/150! overall sequences, that is, more than 500 million combinations.

▪ Before launching the analysis, it was necessary to non-dimensionalize the basic units of measurement to better compare the results for the two propulsions.

4 Electric Propulsion

4.1 History background

The idea of using an electric propulsion based engine on spacecraft can be dated back to 1906 when Robert Goddard considered its possible use for space missions and when Konstantin Tsiolkovsky first formulated the Rocket equation law.

The development of first electric propulsion thrusters date back to 1960s: first experiments of electric thrusters occurred on 1964 with the SERT-1 (Space Electric Rocket Test 1) that proved the possibility of obtaining thrust through an electric thruster (therefore capable of delivering substantial current and applying it for space missions), and with the soviet satellite Zond-2 that mounted a Pulsed Plasma Thruster (PPT) aboard.

Despite these preliminary tests, first electric thrusters were developed only during 90s and were first applied to spacecraft in the following decade, when EP reached its full potential due to the advancing technology and increasing available power on board the spacecrafts.

Figure 15: SERT-1 Figure 16:Zond-2

4.2 Fundamentals of Electric propulsion

Electric propulsion lies on the momentum conservation law: within some problem domain, the momentum remains constant and, therefore, it’s neither created nor destroyed but, through the action of a force (thrust) the momentum can be changed (described by Newton’s law of motion). Through the ejection of matter (e.g. high speed exhausted gas) in a specific direction, the spacecraft will respond by accelerating in the opposite direction (principle of action and reaction).

푑푣 푑푚 푚 = 푐 푑푡 푑푡

Table 2: Momentum conservation law terms

푚 Time varying (instantaneous) mass of the vehicle

푑푣 Spacecraft acceleration

푑푡

푐 Velocity of exhaust stream

푑푚 Rate of change of spaceraft mass

푑푡

푑푚 푇 = 푐 푑푡

Table 3: Thrust terms

푇 Thrust

Figure 17: Rocket engine thrust

The thrust can be assumed as if it was an external force applied to the vehicle: the integral of thrust over time is the impulse (or change of momentum).

The key point for EP that distinguish it from CP, is that it achieves a value of propellant exhaust speed high because a large amount of energy can be transferred from the power source to the propellant.

A low amount of propellant is the main advantage of the EP because a high exhausted velocity directly translates in low amount of propellant consumption.

CP, instead, converts chemical energy from the propellant to kinetic energy and so, propellant consumption is higher.

Electric propulsion does not have energy limitations: by ignoring the components deterioration, the energy provided by solar panels and power source to a small amount of propellant is definitely bigger than the one provided with chemical propulsion.

The only limitation in energy quantity with electric propulsion lies in the solar panels and power source mass. However, the specific impulse (퐼푠푝) for the EP is bigger and, therefore, electric propulsion based engines can provide thrust for longer time (up to few years) using less quantities of propellants.

EP has a big disadvantage that makes it less attractive when the mission length is short: the limitations on power source reduce the available thrust so thrust is defined

as “low”. The consequence is that the operational times are long and that makes it difficult for short-timed missions.

A key term in the study of electric propulsion is the specific impulse: it is the ratio of thrust over the ration of expelled propellant measured in units of weight expelled per second.

푇 푐 퐼푠푝 = = 푊 푔0

Table 4: Specific impulse law terms

퐼푠푝 Specific impulse

푑푚 Rate of expulsion of propellant 푊 = 푔 푑푡 0

푔0 Gravitational acceleration at sea level

Electric propulsion has specific impulse 10 times greater than the specific impulse obtained through the chemical propulsion. The specific impulse can be described as a measure of how efficient a rocket uses propellant; by definition, it is the total impulse delivered per unit of propellant consumed and is dimensionally equivalent to the generated thrust divided by the propellant mass or weight flow rate (thus meaning it has units of velocity).

A propulsion system with a high specific impulse uses the propellant mass way more efficiently in creating thrust and, in case of rockets, less propellant for a given Δ푉. If the exhaust velocity (푐) is assumed constant during thrust, the spacecraft experiments an increment in velocity which is linearly dependent on the exhaust velocity and logarithmically dependent on the propellant mass ejected, because of the Tsiolkovsky rocket equation:

푚 Δ푉 = 푐 ln 0 푚푓

Table 5: Tsiolkovsky rocket equation terms

Δ푉 Increment in velocity

푐 Velocity of exhaust stream

푚0 Initial total mass

푚푓 Final total mass without the propellant

Electric spacecraft propulsion systems create thrust by using electric, and possibly magnetic, processes to accelerate a propellant. More intense forms of propellant heating, as used in electrothermal propulsion systems, offer one possibility for increased exhaust velocity, but encounter limitations due to restrictions on the temperatures that can be sustained by engine components in contact with the propellant gas flow. Thermodynamic expansion can be abandoned in favour of direct application of body forces to particles in the propellant stream. This is the method used by electrostatic and electromagnetic propulsion systems.

4.3 Thrusters classification

The principle of EP consists in applying electrical energy to the propellant from an external power source to obtain a high exhaust speed well above what the chemical propulsion engines produce. EP thrusters are classified in three main categories, according to the way they transfer energy to the propellant and the way that thrust and specific impulse are generated (that is to say how the plasma is accelerated).

- Electrothermal propulsion: the propellant in the chamber is heated thanks to an electrical component (e.g. a resistance) up to a specific temperature. The momentum on the spacecraft is imposed after the heat gas is accelerated through a nozzle where the thermal energy is transformed in kinetic energy. - Electrostatic propulsion: the propellant is first ionized thanks to a power source, then it is accelerated by means of electrodes in the direction of the applied electric field. - Electromagnetic propulsion: the propellant is ionized thanks to a power source, then it is accelerated thanks to the combined action of applied electric field and magnetic one (electric and magnetic forces).

Reistojets - Arcjets

In resistojets propellant in the chamber is heated by means of an electric resistance powered with electric energy. Heat passes from the resistance to the propellant through the mechanism of convection or irradiation. There are two main configurations:

- Direct contact: the resistance is in contact with the flow; heat passes from resistance to propellant through convection mechanism. - Sealed cavity: resistance stays in a sealed cavity. Heat passes from resistance to propellant through irradiation mechanism, avoiding any sort of breaking mechanisms. For this configuration, it is necessary to over-heat the resistance because of the heater walls.

Resistojets have low specific impulses (퐼푠푝) because the temperature of the gas is lower than the temperature of the resistance, high efficiency, limited power so they 31

are used for not very demanding missions (orbit insertion and station keeping), thery are simple, cheap and reliable.

Arcjets have been developed to overcome the limitations on temperature of resistojets (where gas temperature is lower than resistance one). The propellant is first ionized or partially ionized, then the heat is deposited directly in the propellant (that becomes itself a conductor) where the current flow generates an electric arc. The arc heats the propellant in a not uniform way so that close to the propeller walls, the propellant has a lower temperature.

- A potential difference is applied to the ionized gas so that all the electrons reach the same value of current 푗. - A ripple effect establishes a mechanism of collisions between electrons and atoms that results to the ionization of atoms first (because first ionization

energy 휖푖푗 of electrons is greater). It follows that the current grows rapidly with a non-linear trend. - The electric field 퐸 accelerates the ions towards the cathode that emits electron because of the ions bombing.

- Sparkling potential 푉퐶 is reached (up to a thousand Volts): a sudden discharge is born and the current 푗 tends to infinite, so it is necessary to limit it value by decreasing the potential with a luminescence discharge. - The cathode needs to be heaten so it starts to emit ions by thermionic effect. The value of current 푗 increase up to generate an electric arc. - Temperature is about 10.000K (far from propeller walls), number of collisions increases and potential needs to decrease.

Figure 18: Arcjet

Figure 19: Arcjet,푉 푣푠 푙푛 푗

Table 6: Resistojets performance

Propellant 푁2퐻4 푁퐻3

퐼푠푝 [푠] 300 350

푃퐸 [푊] 500-1500 500

휂 0.8 0.8

Voltage 28 28

Thruster mass [kg/KW] 1-2 1-2

PPU mass [kg/KW] 1 1

Feed system Blowdown Regulated

Lifetime [h] 500 -

Missions SK, orbit insertion, Orbit corrections deorbit

Table 7: Arcjets performance

Propellant 푁2퐻4 푁퐻3 퐻2

퐼푠푝 [푠] 500-600 500-800 1000

푃퐸 [푊] 300-2000 500-30k 5k-100k

휂 0.35 0.3 0.4

Voltage 100 100 200

Thruster mass 0.7 0.7 0.5 [kg/KW]

PPU mass 2.5 2.5 2.5 [kg/KW]

Feed system Regulated Regulated Regulated

Lifetime [h] 1000 1500 -

Missions SK SK, orbit raising Medium Δ푉 transfers

Gridded Ion thrusters

The gridded ion thruster is a type of electrostatic thruster capable of generating low thrust levels with very high efficiency. This type of propeller exploits the acceleration of ions in the chamber thanks to the presence of high-voltage electrodes, capable of generating electrostatic forces that push the ions in the axial direction. The first applications date back to the nineties when engines of this type were mounted on board the Deep Space 1 probe.

The operating principle is as follows:

- The propellant (not yet ionized) atoms are sprayed into the chamber where an electronic cannon bombards them causing the electrons to be removed from the propellant atoms. The propellant is then ionized and positively charged, while the propeller walls absorb the lost electrons. - The positively charged ions move to the chamber outlet because of diffusion, they escape into a plasma casing just above the positively charged grid. - The ions are trapped between the two grids (positively and negatively charged respectively). They are accelerated in the direction of the negatively charged grid and then expelled in outer space. - Electrons are fired through the ions by a cathode, called a neutralizer, to ensure that an equal amount of positive and negative charges is expelled. Neutralization is necessary to prevent the ship from gaining a net negative charge.

Figure 20: Diagram of a Gridded Ion Thruster

Hall thusters

Hall thrusters or Stationary plasma thrusters is a hybrid between electrostatic and electromagnetic propulsion. This type of engine has the great advantage of not being subject to the limitation on the thrust density imposed by the Child law:

1.5 4휖0 2푞 푉퐺 푗푚푎푥 = √ 2 9 푚+ 푥푎

which is the distinctive limit of the ion thrusters (because of a close-to-neutral plasma in the chamber and therefore there is no separation of charges).

The Hall effect thrusters base their operating principle on the acceleration of a working fluid (propellant) conveniently ionized by the mutual action of the superposition of a magnetic field and an electric field orthogonal to each other and directed respectively radially and along the axis of the propeller usually realized with cylindrical symmetry.

An electron discharge current is emitted from the cathode such that it flows in an almost axial direction towards the anode. When the electrons penetrate inside the thruster and are affected by the radial magnetic field they remain trapped in the zone of maximum intensity of the magnetic field which practically stops their motion towards the anode due to the small cyclotron radius of the electrons and gives them an azimuth speed creating a circumferential electronic current inside the propeller by Hall effect.

The magnetic force to which the electron is subject is given by:

퐽푒 퐹퐵 = 푞푣푒 × 퐵 = 푞퐸 = × 퐵 푛푒

The electron is subject to a result of the null forces because the electrostatic force equals the magnetic force:

−푞퐸푧 = 푞퐵푣휃

There is only a Hall current 푗휃 equal to:

푗휃 = −푛푒푞푣휃

The discharge is constituted by a high density of high energy electrons and allows ionization of the propellant, generally injected into the thruster through an annular distribution chamber constituted by the same anode, by means of the impact of the electrons with the working atoms. At each collision the electron advances generating a current towards the anode.

The distribution of electrons in the discharge produces a negative spatial discharge effect (virtual cathode) which generates a potential difference with the anode allowing the acceleration of the produced ions. For this reason, a current of ions is generated: the Hall effect thrusters are also defined as non-grid ion motors.

The ions accelerated by the potential difference are not affected by the action of the magnetic field (B) due to both their high atomic mass and their low speed, presenting a large radius of cyclotron and thus traveling along almost straight rectilinear paths mainly along the axis of the motor. The ions are then accelerated by the electromagnetic field in the axial direction.

Figure 21: Hall thruster

Compared to ion propellers, Hall effect ones have a higher thrust density, a much lower ionization cost. They are suitable for missions characterized by specific pulses lower than those of missions for which ion thrusters are typically used.

However, plasma is an unstable environment: in fact, it can cause electromagnetic interference or radio-frequency interference and erosion of the internal surfaces of the engine; the performances can be influenced by the impact of the ions against the walls of the engine, the divergence of the beam, the presence of double ions.

Table 8: Hall thrusters vs Ion thrusters performance

Type 퐻푎푙푙 퐼표푛

Propellant 푋푒 푋푒

퐼푠푝 [푠] 1500-2500 2000-4000

푃퐸 [푊] 300-6000 200-5000

휂 0.5 0.65

Voltage 200-600 1000-2000

Thruster mass 2-3 3-6 [kg/KW]

PPU mass [kg/KW] 6-10 6-10

Feed system Regulated Regulated

Lifetime [h] >7000 >10000

Missions SK, orbit transfer SK, orbit transfer (medium Δ푉) (large Δ푉)

MPD

A magnetoplasmadynamic (MPD) thruster (MPDT) is a form of electrically powered spacecraft propulsion which uses the Lorentz force (the force on a charged particle by an electromagnetic field) to generate thrust. It is sometimes referred to as Lorentz Force Accelerator (LFA) or (mostly in Japan) MPD arcjet (as an evolution of them). NASA refers to it as the “the most powerful form of electromagnetic propulsion”

Generally, a gaseous material is ionized and fed into an acceleration chamber, where the magnetic and electrical fields are created using a power source. The particles are then propelled by the Lorentz force resulting from the interaction between the current flowing through the plasma and the magnetic field (which is either externally applied, or induced by the current) out through the exhaust chamber. Unlike chemical propulsion, there is no combustion of fuel. As with other electric propulsion variations, both specific impulse and thrust increase with power input, while thrust per watt drops.

There are two main types of MPD thrusters, applied-field and self-field. Applied- field thrusters have magnetic rings surrounding the exhaust chamber to produce the magnetic field, while self-field thrusters have a cathode extending through the middle of the chamber. Applied fields are necessary at lower power levels, where self-field configurations are too weak. Various propellants such as xenon, neon, argon, hydrogen, hydrazine, and lithium have been used, with lithium generally being the best performer.

MPD thruster has two metal electrodes: a central rod-shaped cathode and a cylindrical anode surrounding the cathode. As for the arcjet, an electric arc is struck between anode and cathode thanks to high current. As the cathode heats up, it emits electrons that collide with ionized propellant to generate plasma. A magnetic field is created by the electric current returning to the power supply through the cathode, just like the magnetic field that is created when electrical current travels through a wire

According to Edgar Choueiri magnetoplasmadynamic thrusters have input power 100–500 kilowatts, exhaust velocity 15–60 kilometers per second, thrust 2.5–25 newtons and efficiency 40–60 percent.

One potential application of magnetoplasmadynamic thrusters is the main propulsion engine for heavy cargo and piloted space vehicles for long-term missions such as for Moon and Mars exploration, outer planet rendezvous.

Figure 22: MPD thruster

Table 9: PPT vs AF-MPD vs SF-MPD performance

Type PPT AF-MPD SF-MPD

퐼푠푝 [푠] 500-1000 2000-5000 2000-5000

푃퐸 [푊] 1-200 1k-100k 200k-4M

휂 0.1 0.5 0.3

Voltage 1000-2000 200 100

Thruster mass 120 - - [kg/KW]

PPU mass 100 - - [kg/KW]

Lifetime [h] 107 pulse - -

Missions Precise corrections Large Δ푉 (medium Large Δ푉

푃퐸) (large 푃퐸)

VASIMR

The VASIMR (Variable Specific Impulse Magnetoplasma Rocket) is a new type of electromagnetic thruster. Gases like xenon, argon or hydrogen are injected into a tube surrounded by superconducting magnets and a series of two radio wave couplers. The couplers have the function of ionizing the gas and, therefore, convert the cold gas into super heated plasma and the magnetic nozzle, in turn, converts the thermal motion of the plasma into a directional jet. The method by which the VASIMR generates thrust derives from research on nuclear fusion.

The first coupler is the helicon coupler that launches helical waves in the direction of cold gas and ionizes it. At this point the gas is a plasma but at a lower temperature than the operating temperature: for this reason it is called "cold plasma" (even if the temperatures reach 5800K). The second coupler is the ICH (Ion Cyclotron Heating) that allows to heat the plasma up to millions of K by means of waves thrown against the ions as they orbit around the magnetic field resulting in accelerated motion and an increase in temperature.

The motion of the ions takes place in a direction perpendicular to that of motion and it is necessary to modify its direction to generate thrust. This is possible thanks to the magnetic nozzle which converts the motion of the ions into useful linear momentum allowing the ions to reach speeds up to 180000 km / h.

Figure 23: VASIMR

Pulsed Inductive Thrusters

Pulsed Inductive Thruster (PIT) was created to overcome the fundamental problem of electrode erosion in Pulsed Plasma Thrusters. The erosion is due to the high voltage of the formation of the discharge, to the high emission current of the electrodes, due to the bombardment of the ions flow and to the high temperature of the plasma. Specifically, electrode erosion is the factor that limits the useful lifetime of electromagnetic thrusters.

In this type of engine, this problem can be overcome by considering the coupling between the electrical energy impulse coming from the energy storage circuit, and the gas inductively, thus removing the direct physical contact between the electrodes and discharge into the plasma.

Therefore, the non-stationariety is used to obtain currents that vary over time, so as to exploit the phenomenon of mutual induction.

A PIT is basically constituted by a flat spiral induction coil, typically one meter in diameter, on which the gaseous propellant is injected.

Through this circuit a strong current impulse (100kA) is passed which rapidly leads to the growth of a magnetic field which permeates the gaseous propellant. According to Maxwell's law of induction, a corresponding electric field develops, whose module must be high enough to transform the gas into plasm, or in a state of high conductivity.

In turn, the magnetic field generates a secondary current inside the plasma opposite to the external current but in phase with it. The interaction between this secondary current and the magnetic field produces the magnetic force that allows the plasma to accelerate in the axial direction.

It is the current induced by the magnetic field that interacts profitably with the plasma. Each resistive current generated by the electric field is 90 degrees out of phase with respect to the magnetic field, so the corresponding magnetic force component reverses the sign every half cycle.

This engine is a promising system and the prototypes have provided encouraging results for applications such as primary propulsion in interplanetary missions. It has much higher returns than PPTs. The specific impulses are high:

퐼푠푝 = 4000 ÷ 9000 푠

As the accelerated plasma moves away from the external circuit, the effect of mutual induction decreases and therefore the accelerating force decreases in intensity. To solve this problem one could resort to a magnetic field traveling with the propellant. This device can be implemented by using alternately activated pairs of electrodes. A 1 megawatt system could pulse 200 times per second.

Figure 24: Pulsed Inductive Thruster (PIT)

Figure 25: Propellant Injection Schematic

4.4 Propellants

Table 10: List of typical propellants for EP

Pros Cons

퐻2 Low molecular mass ℳ it dissociates easily, it is cryogenic and therefore difficult to preserve.

퐻푒 Low molecular mass ℳ It is cryogenic and therefore difficult to preserve

퐿푖 Low molecular mass ℳ Toxic

푁퐻3 Liquid at room temperature, Low It dissociates easily. molecular mass ℳ after dissociation. Chemically reactive and Easy to store. requires vaporizer.

푁2퐻4 Liquid at room temperature, Low It dissociates easily and needs molecular mass ℳ after dissociation. a gas generator. Easy to store. exothermic dissociation

퐻2푂 Easily storable. Great availability. Non- It dissociates easily, needs a toxic and non-polluting. vaporizer and condensation must be avoided.

5 Results

The results for the space mission scenario about multiple debris removal with electric propulsion are here presented: the key point in the problem is the choice of electric propulsion onboard the chaser satellite instead of the chemical one.

Following the procedure summarized here, rendezvous times are first determined by studying the time at which the orbit planes of the debris coincide.

The procedure resulted in 109 combinations for acceleration values of 10−5 푚/푠2 and 10−6 푚/푠2 and 58 combinations for an acceleration equal to 10−7 푚/푠2 with time length shorter than 6 months, shown in Figure 9, Figure 10 and Figure 11.

The code has discarded regardless of those combinations that provided the chaser to return two or more times to the same debris. In addition, the code has rejected those combinations of debris that provided a time of rendezvous between one debris and another lower than the thrust time.

This is because it would be impossible to reach a debris with electric propulsion with thrust time greater than transfer time.

Results for the J2 perturbations on the orbital element Ω for the debris sequence “27819 – 25893 – 8326 – 11309” show how the RAAN is perturbed daily of:

NORAD No. Daily perturbation of Ω [°/day]

27819 -0.0984

25893 -0.0887

8326 -0.0072

11309 -0.1150

Figure 26:휴 perturbation for the debris combination over the mission time

This will directly influence the time for rendezvous and proximity operations. By imposing 푡 = 0 푑푎푦푠 the departure time from the debris and ΔΩ = Ω푖 − Ω푗 = 0 (between the departure debris and the arrival one), the condition for performing a correct rendezvous, the thrust time is finite and should be properly chosen to catch the debris at the right time and place.

For each of the 109 (or 58) combinations, time-related data, consumption data (in terms of Δ푉 of individual legs and complete transfer) and data related to the de-skid planes offset were printed on output files.

Among all the provided solutions, the same sequences were analysed for electric propulsion and for chemical propulsion, evaluating the differences in terms of mission time and consumption, propellant consumption and solar arrays mass (for the three types of EP thrusters only).

5.1 Output tables with dimensional results

In the following pages, there will be presented the results according to the analysis of a space mission about multiple debris removal with an electric propulsion based chaser.

Input data for the analysis started by imposing multiple acceleration values to satisfy the different thrust on mass ratios for different propulsion engines.

Table 11: Thrust on mass ratios (Acceleration)

Acceleration [풎/풔ퟐ]

1 ⋅ 10−6

1 ⋅ 10−7

1 ⋅ 10−5

The mass of the spacecraft is the sum of the mass of the chaser, that of the kit and the mass of the solar panels.

Starting from the necessary output power, technology level and efficiency the mass for the solar panels has been calculated as follow:

푇 ⋅ 퐼푠푝 ⋅ 푔0 푃 = 2휂

(훽 ⋅ 푇 ⋅ 퐼푠푝 ⋅ 푔0) 푚 = = 훽 ⋅ 푃 푠표푙푎푟 푝푎푛푒푙푠 2휂 where

- 훽 = 30 푘푔/푘푊

- 퐼푠푝 = 3000 푠 (퐼표푛), 1800 푠 (퐻푎푙푙), 600 푠 (퐴푟푐푗푒푡) 2 - 푔0 = 9.8 푚/푠 - 푎 = 1 ⋅ 10−6 푚/푠2, 1 ⋅ 10−7 푚/푠2, 1 ⋅ 10−5 푚/푠2

- 휂 = 0.65 (퐼표푛), 0.5 (퐻푎푙푙), 0.35 (퐴푟푐푗푒푡)

푚푆퐶 = 푚푐ℎ푎푠푒푟 + 푚푘푖푡 + 푚푠표푙푎푟 푝푎푛푒푙푠 ≅ 2500 푘푔

This result for the mass of the spacecraft is an approximation due to the unknown value of the solar panel mass and it’s been the starting point for the mass budget calculation (specific results are presented in the following pages).

The mass of the solar panels considers only the chaser, because, as already mentioned, the kit is equipped with chemical propulsion.

The deorbit phase is supposed with chemical propulsion: deorbit with electric propulsion is more economical in terms of Δ푉, but would provide considerable mission time and human resources in terms of mission control operators as well as the exact calculation of the point of re-entry into the atmosphere remains an unknown factor of the problem.

Table 12: Solar panel mass and Power for 푎 = 1 ⋅ 10−5 푚/푠2

Engine type Solar panel mass [풌품] Power [푾]

Ion 17 565

Hall 13.23 441

Arcjet 6.3 210

Table 13: Solar panel mass and Power for 푎 = 1 ⋅ 10−6 푚/푠2

Engine type Solar panel mass [풌품] Power [푾]

Ion 1.7 56.5

Hall 1.32 44

Arcjet 0.63 21

Table 14: Solar panel mass and Power for 푎 = 1 ⋅ 10−7 푚/푠2

Engine type Solar panel mass [풌품] Power [푾]

Ion 0.17 5.6

Hall 0.13 4.5

Arcjet 0.06 2

The power generated by the ion thrusters, in the 3 different scenarios, is bigger, as the specific impulse (퐼푠푝) is greater than the Hall thrusters and Arcjets, and so the mass of the solar panels for ion thruster is bigger, taking into account the same technological level (훽) for all the three thrusters.

Each of the output files was organized on a calculation page on Excel and all the combinations of four debris were compared.

For all the three acceleration estimation, the results in terms of minimum consumption and maximum consumption are the same; the inequality was found in the thrust time for the multiple transfers.

This is obviously a direct consequence of the divergence in the acceleration quantities.

- For the acceleration value of 1 ⋅ 10−6 푚/푠2 thrust time requires few days (2 to 5 days) for burning. - For the acceleration value of 1 ⋅ 10−7 푚/푠2 thrust time requires tens of days (12 to 44 days) for burning. - For the acceleration value of 1 ⋅ 10−5 푚/푠2 the thrust time is significantly lower, that is to say few hours (less than an hour to 10 hours) for burning.

Table 15: Mean thrust time for different acceleration values

Acceleration [풎/풔ퟐ] Mean thrust time [days]

10−6 3.012

10−5 0.301

10−7 28.502

The electric propulsion transfers with the lowest acceleration value of 10−7푚/푠2 appear to be problematic as the mean evaluated thrust time is greater, compared to the other two studied cases (푎 = 1 ⋅ 10−6 푚/푠2 and 푎 = 1 ⋅ 10−7 푚/푠2).

This parameter needs to be compared with the time necessary for the orbit planes to coincide, that is to say the time necessary for the rendezvous.

For this reason, the four-debris combinations with that specific acceleration quantity (푎 = 1 ⋅ 10−7 푚/푠2) are less in number than the other two cases.

The total consumption (in terms of Δ푉) for the entire mission is the same for the three studied cases considering that the Δ푉 does not depend on the acceleration of the spacecraft but only on the perturbed variation of semi-major axis and orbit eccentricity.

Table 16: Best Three-Leg Missions in Terms of 휟푽 for 풂 = ퟏ풆 − ퟎퟔ 풎/풔ퟐ

Sequence NORAD No. Trip time, days 횫V, km/s

#1 #2 #3 #4

1 27819 25893 8326 11309 194.620 0.21866

2 27466 27535 11170 10777 183.735 0.45444

3 13066 11170 27535 28421 151.769 0.45709

4 27535 27466 13003 28910 192.889 0.48999

5 27466 27535 11170 28421 191.091 0.50248

Table 17: Best Three-Leg Missions in Terms of 휟푽 for 풂 = ퟏ풆 − ퟎퟓ 풎/풔ퟐ

Sequence NORAD No. Trip time, days 횫V, km/s

#1 #2 #3 #4

1 27819 25893 8326 11309 194.620 0.21866

2 27466 27535 11170 10777 183.735 0.45444

3 13066 11170 27535 28421 151.769 0.45709

4 27535 27466 13003 28910 192.889 0.48999

5 27466 27535 11170 28421 191.091 0.50248

Table 18: Best Three-Leg Missions in Terms of 휟푽 for 풂 = ퟏ풆 − ퟎퟕ 풎/풔ퟐ

Sequence NORAD No. Trip time, days 횫V, km/s

#1 #2 #3 #4

1 27819 25893 8326 11309 205.2096 0.21866

2 27466 27535 11170 10777 183.735 0.45444

3 13066 11170 27535 28421 173.1984 0.45709

4 27466 27535 11170 28421 191.0910 0.50248

5 16511 27870 11170 10777 163.735 0.50714

Table 19: Best Three-Leg Missions in Terms of Trip Time 휟풕 for 풂 = ퟏ풆 − ퟎퟔ 풎/풔ퟐ

Sequence NORAD No. Trip time, days 횫V, km/s

#1 #2 #3 #4

1 13066 13003 27535 11170 123.8727 0.84296

2 27535 27466 13003 21090 125.15942 0.73455

3 13066 13003 27535 27870 126.37418 0.99733

4 13003 13066 27870 27535 126.37418 0.88459

5 11170 13066 27870 27535 126.37418 0.61615

Table 20: Best Three-Leg Missions in Terms of Trip Time 휟풕 for 풂 = ퟏ풆 − ퟎퟓ 풎/풔ퟐ

Sequence NORAD No. Trip time, days 횫V, km/s

#1 #2 #3 #4

1 13066 13003 27535 11170 123.8727 0.84296

2 27535 27466 13003 21090 125.15942 0.73455

3 13066 13003 27535 27870 126.37418 0.99733

4 13003 13066 27870 27535 126.37418 0.88459

5 11170 13066 27870 27535 126.37418 0.61615

Table 21: Best Three-Leg Missions in Terms of Trip Time 휟풕 for 풂 = ퟏ풆 − ퟎퟕ 풎/풔ퟐ

Sequence NORAD No. Trip time, days 횫V, km/s

#1 #2 #3 #4

1 11170 13066 27870 27535 151.31328 0.61615

2 11170 13066 28421 27535 153.58982 0.61779

3 11170 13066 27870 10777 162.34808 0.62282

4 27870 16511 28421 27535 168.48352 0.76246

5 27535 27466 10777 27870 169.16294 0.77037

Interesting results can be found in the “best three-leg missions in terms of mission time” tables for the different acceleration values: in fact, the first four best

sequences for the acceleration equal to 1 ⋅ 10−6 푚/푠2 and 1 ⋅ 10−5 푚/푠2 (13066, 13003, 27535, 11170 – 27535, 27466, 13003, 21090 – 13066, 13003, 27535, 27870 – 13003, 13066, 27870, 27535) does not appear in the best five sequences for the acceleration value of 1 ⋅ 10−7 푚/푠2 where the best sequence in terms of trip time (11170,13066,27870,27535) is equal to the fifth best combination of four debris in the other two context.

Table 22: Minimum delta V for Electric Propulsion based engines

Acceleration [풎/풔ퟐ] Minimum 횫푽 budget [km/s]

10−6 0.21866

10−5 0.21866

10−7 0.21866

Table 23: Maximum delta V for Electric Propulsion based engines

Acceleration [풎/풔ퟐ] Maximum 횫푽 budget [km/s]

10−6 1.01078

10−5 1.01078

10−7 1.01078

The four-debris combinations that provide the minimum or maximum consumption for the space mission are, obviously, the same:

Table 24: Four-debris combination providing minimum delta V

27819 25893 8326 11309

Table 25: Four-debris combination providing maximum delta V

13066 13003 12092 27466

- 27819 is a SL-8 Rocket body, launched on June 4, 2003 from Plesetsk Missile and Space Complex (PKMTR)

- 25893 is a SL-8 Rocket body, launched on August 26, 1999 from Plesetsk Missile and Space Complex (PKMTR)

- 8326 is a SL-8 Rocket body, launched on September 24, 1975 from Plesetsk Missile and Space Complex (PKMTR)

- 11309 is a SL-8 Rocket body, launched on March 21, 1979 from Plesetsk Missile and Space Complex (PKMTR)

- 13066 is a SL-8 Rocket body, launched on February 17, 1982 from Plesetsk Missile and Space Complex (PKMTR)

- 13003 is a SL-8 Rocket body, launched on December 17, 1981 from Plesetsk Missile and Space Complex (PKMTR)

- 12092 is a SL-8 Rocket body, launched on December 10, 1980 from Plesetsk Missile and Space Complex (PKMTR)

- 27466 is a SL-8 Rocket body, launched on July 8, 2002 from Plesetsk Missile and Space Complex (PKMTR)

Figure 27: 2D Graphic (Ground track) of the four-debris

Figure 28: 3D Graphic (Orbit View) of the four-debris

The four-debris combinations that provide the minimum mission time are, contrarily, different for the three quantities of acceleration.

Table 26: Four-debris combination providing minimum mission time

Acceleration [푚/푠2]

10−6 13066 13003 27535 11170

10−5 13066 13003 27235 11170

10−7 11170 13066 27870 27535

The minimum mission time with electric propulsion is the same for the acceleration values of 10−6 푚/푠2 and 10−7 푚/푠2, considering that the 4-debris combination is the same, while the minimum mission time for the lower acceleration value of 10−7 푚/푠2 appears to be greater because of the different debris combination.

Table 27: Minimum mission time for Electric propulsion based engines

Acceleration [풎/풔ퟐ] Minimum mission time [days]

10−6 123.8727

10−5 123.8727

10−7 151.31328

5.2 Mass budget

The table below shows the breakdown of the mass budget, providing the values for mission initial mass 푚푠푡푎푟푡, kit mass 푚푘 and propellant consumed during each leg

푚푝 for the sequences with trip time below 6 months (in terms of minimum and maximum). Values for power (P) have been considered only for electric propulsion cases.

The calculation of the masses for the space mission follows the assumption that the kit equipped with chemical propulsion based thruster employs a liquid propulsion system with specific impulse 퐼푠푝 = 310 푠.

Mass budget breakdown has been calculated varying the specific impulses for the three type of electric thrusters for the three values of acceleration (10−5 푚/ 푠2, 10−6 푚/푠2, 10−7 푚/푠2).

This is to say that only the chaser spacecraft mounts an electric propulsed thruster onboard while all the kits are equipped to perform thrust with chemical propulsion.

An ion thruster (퐼푠푝 = 3000푠), a Hall thruster (퐼푠푝 = 1800푠) and arcjet (퐼푠푝 = 600푠) are here considered.

Table 28: Mass budget breakdown for 풂 = ퟏ ⋅ ퟏퟎ−ퟓ 풎/풔ퟐ

푪풉풂풔풆풓 푰풔풑 [풔] 풎풔풕풂풓풕 [풌품] 풎풌 [풌품] 풎풑 [풌품]

min max min max min max

CP 310 2166 3817 164 275 9 423

Ion 3000 2715 2971 164 275 1 35 thruster

Hall 1800 2727 3015 164 275 2 60 thruster

Arcjet 600 2784 3314 164 275 6 194

Table 29: Mass budget breakdown for 풂 = ퟏ ⋅ ퟏퟎ−ퟔ 풎/풔ퟐ

푪풉풂풔풆풓 푰풔풑 [풔] 풎풔풕풂풓풕 [풌품] 풎풌 [풌품] 풎풑 [풌품]

min max min max min max

CP 310 2166 3817 164 275 9 423

Ion thruster 3000 2715 2971 164 275 1 35

Hall thruster 1800 2727 3015 164 275 2 60

Arcjet 600 2784 3314 164 275 6 194

Table 30: Mass budget breakdown for 풂 = ퟏ ⋅ ퟏퟎ−ퟕ 풎/풔ퟐ

푪풉풂풔풆풓 푰풔풑 [풔] 풎풔풕풂풓풕 [풌품] 풎풌 [풌품] 풎풑 [풌품]

min max min max min max

CP 310 2166 3817 164 275 9 423

Ion thruster 3000 2715 3314 164 275 1 194

Hall thruster 1800 2727 3314 164 275 2 194

Arcjet 600 2784 3314 164 275 6 194

5.3 Electric propulsion – Chemical propulsion comparison

There are two main differences between the electric propulsion and the chemical propulsion: the first one is the type of burn the spacecraft performs to make a manoeuvre because the thrust with the chemical propulsion needs an impulsive burn as the propellant requires it, while the thrust with the electric propulsion is performed with a finite burn considering that the spacecraft mounts a power source (a generator); the second difference lies in the estimation of the optimal transfer Δ푉 that can be expressed as:

Δ푉 Δ푎 2 = 0.5√( ) + Δ푒2 푉 푎 where the change in semimajor axis Δ푎 is considered and empirical relation and Δ푒 is the additional eccentricity vector change.

Table 31: Comparison between EP and CP in terms of 휟푽

Minimum Δ푉 Maximum Δ푉

Chemical propulsion 0.21539 1.0076

Electric propulsion 0.21866 1.01078

Table 32: Comparison between EP and CP in terms of 풕

Minimum 푡 Maximum t

Chemical propulsion 103.8727 182.62815

Electric propulsion 103.8727 182.62815

Electric propulsion requires a slightly higher value for Δ푉 [푘푚/푠] compared to that of the chemical propulsion: specifically, 1.5% more Δ푉 is required.

2900

2800

2700

2600

2500

2400 Spacecraft Spacecraft mass [kg] 2300

2200

2100 1 2 3 4

Ion Hall Arcjet CP

Figure 29: Spacecraft mass decrease during the best three-leg mission for CP and EP thruster, for all the acceleration values

In all three cases of acceleration, for the different types of engine, the mass of the spacecraft decreases with each leg because part of it is used in the kit that is released on the debris and part is in the propellant consumed to perform the rendezvous manoeuvres.

The ion thruster solution provides a lower initial mass for the spacecraft (2715.15 kg) compared to the other three cases of Hall thruster (2726.57 kg), arcjet (2784.52 kg) and chemical propulsion thruster (2865.37 kg).

30 Propellant mass [kg]

0 1 2 3

Ion Hall Arcjet CP

Figure 30: Propellant mass consumption for the best three-leg mission for CP and EP thruster, for all the acceleration values

From the graphs shown above, regarding the propellant consumption (in terms of kilograms of propellant necessary for the rendezvous manoeuvre), it is soon clear how the results for EP are considerably better than CP ones.

The high specific impulse (퐼푠푝) of the electric thrusters in general allows up to 10 times saving of propellant.

Taking a look to the second of the three rendezvous manoeuvres, the propellant consumption evaluated for the chemical propulsion is 81.5 kg while, taking into account the best scenario with ion thrusters (greater 퐼푠푝), the propellant consumption is just 8.27 kg for all the three different acceleration values.

Figure 31: Power depending on Solar panel mass for the three electric thrusters

The Figure 31 represents an evaluation of the direct correlation between the necessary onboard power (P) and the required solar panel mass (푚푝).

An almost linear trend is immediately visible: as the required onboard power increases, there is an increase in the mass of the solar panel.

Figure 32: Propellant mass depending on Solar panel mass for the three electric thrusters

Figure 32, shows the trend of the mass of solar panel as a function of the propellant mass: a linear decreasing trend is immediately clear. As the mass of propellant increases, there is a decrease in the mass of the solar panels so that the mass of the spacecraft (which takes into account both masses) remains more or less constant.

Furthermore, an interesting comparison stays in the effective weight saving for the spacecraft equipped with electric propulsion, taking into account the gain in mass because of the propellant and the loss in weight due to the presence of solar panels.

The mass saving, for the EP thrusters, has been calculated as follow:

푚푝 − (푚푝 + 푚푠푝) 푚푎푠푠 푠푎푣푖푛푔 = 퐶푃 ⋅ 100 푚 푝퐶푃

Table 33: Propellant mass and solar panel mass for EP, 푎 = 1 ⋅ 10−5 푚/푠2

Leg 1 Leg 2 Leg 3 푚푝 saving Solar Mass [kg] panel saving

mass [%] 푚푝 [푘푔] [kg]

CP 9.4623 83.28152 74.50974 --

Ion 1.16454 8.27016 7.59486 150.224 17 79.6 thruster

Hall 1.9493 13.82912 12.67288 138.802 13.23 75 thruster

Arcjet 5.97564 42.17735 38.2414 80.859 6.3 44.5

Table 34: Propellant mass and solar panel mass for EP, 푎 = 1 ⋅ 10−6 푚/푠2

Leg 1 Leg 2 Leg 3 푚푝 Solar Mass saving panel saving

[kg] mass [%] 푚푝 [푘푔] [kg]

CP 9.4623 83.28152 74.50974 --

Ion 1.16454 8.27016 7.59486 150.224 1.7 88.8 thruster

Hall 1.9493 13.82912 12.67288 138.802 1.32 82.2 thruster

Arcjet 5.97564 42.17735 38.2414 80.859 0.63 48

Table 35: Propellant mass and solar panel mass for EP, 푎 = 1 ⋅ 10−7 푚/푠2

Leg 1 Leg 2 Leg 3 푚푝 Solar Mass saving panel saving

[kg] mass [%] 푚푝 [푘푔] [kg]

CP 9.4623 83.28152 74.50974 --

Ion 1.16454 8.27016 7.59486 150.224 0.17 89.7 thruster

Hall 1.9493 13.82912 12.67288 138.802 0.13 83 thruster

Arcjet 5.97564 42.17735 38.2414 80.859 0.063 48.3

With the use of electric propulsion onboard our spacecraft, we obtain benefits in terms of mass: in the face of mass spending on the installation of solar panels, we see a fuel saving that is so effective as to completely compensate the additional mass for solar panels.

With the use of ion thruster, the propellant gain reaches up to 90% while for the Hall thruster (the "worst case" for the EP) the gain reaches “just” the 48%.

The possibility of carrying out the mission with electric propulsion – with the same mission time – gives advantages both about the initial mass of the spacecraft and the mass of propellant to be carried on board.

6 References

Edelbaum, Theodore N. "Propulsion requirements for controllable satellites." ARS Journal 31.8 (1961): 1079-1089.

Bate, Roger R., Donald D. Mueller, and Jerry E. White. Fundamentals of astrodynamics. Courier Corporation, 1971.

CASALINO, Lorenzo; PASTRONE, Dario Giuseppe. Mission Design and Disposal Methods Comparison for the Removal of Multiple Debris. 2016.

RYCROFT, Michael J. Stengel; ROBERT, F. Spacecraft Dynamics and Control. 1997. https://celestrak.com http://sci.esa.int/smart-1/34201-electric-spacecraft-propulsion/?fbodylongid=1535 https://www.nasa.gov/centers/glenn/about/fs22grc.html http://www.adastrarocket.com/aarc/VASIMR http://www2.ee.ic.ac.uk/derek.low08/yr2proj/ion.htm http://www.aerospacengineering.net/?p=537 https://www.nasa.gov/mission_pages/station/news/orbital_debris.html

7 Acknowledgments

I’d like to thank professor Lorenzo Casalino first, who guided me during this work, was always present and available for any clarification and in successfully indicating the way for the accomplishment of this thesis.

My thanks also go to the people who – in these years – have supported me, above all from afar, people who have been the lighthouse during the most difficult moments of my studies: my love Simona who loved and supported me in these years, who has always believed in me and has continuously encouraged me to reach this goal as soon as possible.

To my parents Mariantonietta and Pierluigi and my sister Ilaria, who contributed to my well-being, my tranquillity and my happiness here in Turin.

To my grandmother Maria who has faced for the second time a long trip, to personally attend my master thesis session, and to all my uncles and cousins who have decided, for a day, to abandon their daily duties and be present for this very important moment.

To all my friends, to the Cubesat team and all the professors and collaborators I have met during these here, thanks for giving me something useful for the success of this journey.

Farewell Politecnico, see you soon.